Let $f\left(x\right)=\int \frac{7{x}^{10}+9{x}^{8}}{{(1+{x}^{2}+2{x}^{9})}^{2}}dx,x0$, $\lim _{x\to 0}f\left(x\right)=0$ and $f\left(1\right)=\frac{1}{4}$. If $A=\left[\begin{matrix}0 & 0 & 1 \ \frac{1}{4} & {f}^{'}(1) & 1 \ {\alpha }^{2} & 4 & 1\end{matrix}\right]$ and $B=adj(adjA)$ be such that $|B|=81$, then ${\alpha }^{2}$ is equal to [JEE Main 2026, 24 Jan (Shift 2)]

Question

Let $f\left(x\right)=\int \frac{7{x}^{10}+9{x}^{8}}{{(1+{x}^{2}+2{x}^{9})}^{2}}dx,x>0$, $\lim _{x\to 0}f\left(x\right)=0$ and $f\left(1\right)=\frac{1}{4}$. If $A=\left[\begin{matrix}0 & 0 & 1 \ \frac{1}{4} & {f}^{'}(1) & 1 \ {\alpha }^{2} & 4 & 1\end{matrix}\right]$ and $B=adj(adjA)$ be such that $|B|=81$, then ${\alpha }^{2}$ is equal to [JEE Main 2026, 24 Jan (Shift 2)]

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Let \(f\left(x\right)=\int \frac{7{x}^{10}+9{x}^{8}}{{(1+{x}^{2}+2{x}^{9})}^{2}}dx,x>0\), \(\lim _{x\to 0}f\left(x\right…

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